This thesis is an attempt in the direction to formulate methods to the study of the first passage across two barriers. It deals with the 'First passage problems across moving Boundaries'. The compensation technique with the method of images employed in electrodynamics is studied for comparison. A finite dam model with constant release; The first passage time for poisson jump process to escape out of a finite interval; Poisson jump process with possitive increments, with the release taken to be proportionate to the dam level; are some of the aspects considered for this study.

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